How do you determine the third Taylor polynomial of the given function at x = 0 #f(x)= e^(-x/2)#?

Answer 1

#e^(-x/2) = sum_(n = 0)^oo ((-1)^(n)(x^n))/((2^n)n!) = 1 - x/2 + x^2/(4(2!)) - x^3/(8(3!))...#

We know the Maclaurin Series (taylor series centered at 0) for #e^x#.
#e^x = 1 + x + x^2/(2!) + x^3/(3!) + ... + x^n/(n!)#
If you substitute #-x/2# for all of the x's in the series for #e^x#, you get:
#e^(-x/2) = 1 - x/2 + (-x/2)^2/(2!) + (-x/2)^3/(3!) ...#
#e^(-x/2) = 1 - x/2 + x^2/(4(2!)) - x^3/(8(3!))...#

This can be modelled by

#sum_(n = 0)^oo ((-1)^(n)(x^n))/((2^n)n!)#

Hopefully this helps!

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Answer 2

To determine the third Taylor polynomial of the function ( f(x) = e^{-x/2} ) at ( x = 0 ), we follow these steps:

  1. Find the function's derivatives up to the third derivative.
  2. Evaluate these derivatives at ( x = 0 ).
  3. Use these values to construct the third Taylor polynomial.

The derivatives of ( f(x) = e^{-x/2} ) are: [ f'(x) = -\frac{1}{2} e^{-x/2}, ] [ f''(x) = \frac{1}{4} e^{-x/2}, ] [ f'''(x) = -\frac{1}{8} e^{-x/2}. ]

Evaluate these derivatives at ( x = 0 ) to find: [ f(0) = 1, ] [ f'(0) = -\frac{1}{2}, ] [ f''(0) = \frac{1}{4}, ] [ f'''(0) = -\frac{1}{8}. ]

Now, construct the third Taylor polynomial using these values: [ P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3. ]

Substituting the values, we get: [ P_3(x) = 1 - \frac{1}{2}x + \frac{1}{8}x^2 - \frac{1}{48}x^3. ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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